The coordinate ring - Algebraic geometry

[andlook]

I'm looking for help understanding the coordinate ring. The definition I have roughly says

There is a surjection PI:k[t_1 , ... , t_n] ---> k[X] given by restricting a polynomial to X, where X is an algebraic set.

Then k[X] is called the coordinate ring.


As i understand it all this is saying is take any polynomial in k[t_1 , ... , t_n] can now only be evaluated with elements in X. So its like changing the domain of the polynomials, (this change is just a shrinking of the bigger n-dimensional space hence the word restriction)

If my understanding is right, then why is it called the coordinate ring? does this name have a special meaning/derivation?

 

[pat_connell]

The name coordinate ring is derived from the idea that the polynomials in k[t_1 , ... , t_n] can be thought of as functions that are defined over the coordinates of the algebraic set X. By restricting the domain of these functions to X, we get a new ring k[X] which is called the coordinate ring. This ring contains functions which are only defined over the coordinates of X.